TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 12, Pages 5029-5040 S 0002-9947(01)02705-2 Article electronically published on May 22, 2001 THE HIT PROBLEM FOR THE DICKSON ALGEBRA NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM Dedicated to Professor Franklin P. Peterson on the occasion of his 70th bi* *rthday Abstract.Let the mod 2 Steenrod algebra, A, and the general linear group, GL(k, F2), act on Pk := F2[x1, ..., xk] with |xi| = 1 in the usual manne* *r. We prove the conjecture of the first-named author in Spherical classes a* *nd the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra Dk * *:= (Pk)GL(k,F2)is A-decomposable in Pk for arbitrary k > 2. This conjecture* * was shown to be equivalent to a weak algebraic version of the classical conj* *ecture on spherical classes, which states that the only spherical classes in Q0* *S0 are the elements of Hopf invariant one and those of Kervaire invariant one. 1.Introduction Let Pk := F2[x1, . .,.xk] be the polynomial algebra over (the field of two el- ements) F2 in k variables, each of degree 1. The general linear group GLk := GL(k, F2) acts on Pk in the usual manner. Dickson proves in [1] that the ring of invariants, Dk := (Pk)GLk, is also a polynomial algebra Dk ~=F2[Qk,k-1, . .,.Qk* *,0], where Qk,sdenotes the Dickson invariant of degree 2k - 2s. It can be defined by the inductive formula Qk,s= Q2k-1,s-1+ Vk . Qk-1,s, where, by convention, Qk,k= 1, Qk,s= 0 for s < 0 and Y Vk = (~1x1 + . .+.~k-1xk-1 + xk). ~j2F2 Let A be the mod 2 Steenrod algebra. The usual action of A on Pk commutes with that of GLk. So Dk is an A-module. One of the authors has been interested in the homomorphism jk : F2 (Pk)GLk ! (F2 Pk)GLk, A A which is induced by the identity map on Pk (see [3]). Observing that j1 is an isomorphism and j2 is a monomorphism, he sets up the following Conjecture 1.1 (Nguy^e~n H. V. Hu'ng [3]).jk = 0 in positive degrees for k > 2. Let D+kand A+ denote respectively the submodules of Dk and A consisting of all elements of positive degree. Then Conjecture 1.1 is equivalent to D+k A+ .* * Pk ____________ Received by the editors September 29, 1999 and, in revised form, February 22* *, 2000. 2000 Mathematics Subject Classification. Primary 55S10; Secondary 55P47, 55Q* *45, 55T15. Key words and phrases. Steenrod algebra, invariant theory, Dickson algebra. This work was supported in part by the National Research Project, No. 1.4.2. cO2001 American Mathematical Socie* *ty 5029 5030 NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM for k > 2 (see [3]). In other words, it predicts that every GLk-invariant eleme* *nt of positive degree is hit by the Steenrod algebra acting on Pk for k > 2. Conjecture 1.1 is related to the hit problem of determination of F2 Pk. This A problem has first been studied by F. Peterson [9], R. Wood [14], W. Singer [12]* *, and S. Priddy [10], who show its relationships to several classical problems in cob* *ordism theory, modular representation theory, Adams spectral sequence for the stable h* *o- motopy of spheres, and stable homotopy type of classifying spaces of finite gro* *ups. The tensor product F2 Pk has explicitly been computed for k 3. The cases A k = 1 and 2 are not difficult, while the case k = 3 is complicated and was solv* *ed by M. Kameko [8]. It seems unlikely that a very explicit description of F2 Pk A for general k will appear in the near future. There is also another approach, t* *he qualitative one, to the problem. By this we mean giving conditions on elements of Pk to show that they go to zero in F2 Pk, i.e. belong to A+ . Pk. Peterson's A conjecture, which was established by Wood [14], claims that F2 Pk = 0 in degree A d such that ff(d + k) > k. Here ff(n) denotes the number of ones in the dyadic expansion of n. Recently, W. Singer, K. Monks, and J. Silverman have refined the method of R. Wood to show that many more monomials in Pk are in A+ . Pk. (See Silverman [11] and its references.) Conjecture 1.1 presents a large family, wh* *ose elements are predicted to be in A+ . Pk. In [3], one of the authors proves the equivalence of Conjecture 1.1 and a weak algebraic version of the conjecture on spherical classes stating that: There ar* *e no spherical classes in Q0S0 except the elements of Hopf invariant one and those of Kervaire invariant one. He also gives two proofs of Conjecture 1.1 for the case k = 3. In this paper, we establish this conjecture for every k > 2. That Conjec* *ture 1.1 is no longer valid for k = 1 and 2 is respectively an exposition of the exi* *stence of Hopf invariant one classes and Kervaire invariant one classes. We have Main Theorem. D+k A+ . Pk for k > 2. Recently, F. Peterson and R. Wood privately informed us that they had proved the theorem for k = 4 and probably for k = 5. The readers are referred to [4] a* *nd [5] for some problems, which are closely related to the main theorem. Additiona* *lly, the problem of determination of F2 Dk and its applications have been studied by A Hu'ng and Peterson [6], [7]. The paper contains five sections. Section 2 is a preparation on the action of* * the Steenrod squares on the Dickson algebra. We prove the main theorem in Section 3 by means of two lemmata, which are later shown in Section 4 and Section 5 respectively. 2.Preliminaries The action of the Steenrod squares on Dk is explicitly described as follows. Theorem 2.1 ([2]). 8 s r >> Qk,rQk,t2 for i = 2k s >:Qk,s for i = 2 - 2 , 0 otherwise. THE HIT PROBLEM FOR THE DICKSON ALGEBRA 5031 From now on, we denote Qk,sby Qs for brevity. We get æ s-1 Sqa(Qs) = Qs-10 ififa =02< a,< 2s-1 or 2s a < 2k-1 for 0 s < k. Combining this with the Cartan formula, one obtains Corollary 2.2. (a) Sqa(QsR) = QsSqa(R) if 0 < a < 2s-1, (b) Sqa(Q0R) = Q0Sqa(R) if 0 < a < 2k-1 for any polynomial R 2 Pk. Let In (n 0) be the right ideal of A generated by the operations Sq2i for i = 0, . .,.n. Definition 2.3.Suppose R1, R2 2 Pk. Then we write R1 R2 (mod In) if R1+R2 belongs to In . Pk. By convention, R1 R2 (mod In) means R1 = R2 for n < 0. This is an equivalence relation. Lemma 2.4. (a) Sq1(R1)R2 R1Sq1(R2) (mod I0), (b) Sq2(R1)R2 R1Sq2(R2) (mod I1) for any polynomials R1, R2 2 Pk. Proof.(a) From the Cartan formula Sq1(R1)R2+R1Sq1(R2) = Sq1(R1R2), we get (a) by Definition 2.3. (b) We have Sq2(R1R2) = Sq2(R1)R2 + Sq1(R1)Sq1(R2) + R1Sq2(R2) (by the Cartan formula) Sq2(R1)R2 + R1Sq1Sq1(R2) + R1Sq2(R2) (mod I0) (by Part (a)) Sq2(R1)R2 + R1Sq2(R2) (mod I0) (since Sq1Sq1 = 0). Hence, Sq2(R1)R2 + R1Sq2(R2) 2 I1 . Pk and (b) follows. |___| Lemma 2.5. Let R 2 Pk (k 1). If Sq1(R) = 0 and all the monomials of R are of positive degree, then R 0 (mod I0). Proof.The lemma is proved by induction on k. For k = 1, it is easy to see that all the monomials of R are of even degree. Since x2n1= Sq1(x2n-11) for n > 0, the lemma is proved. Let k > 1 and suppose inductively that the lemma holds for polynomials in k - 1 variables. Let us write X R = xi1Ri 0 i 2n for some positive integer n and some polynomials Ri (0 i 2n) in k -1 variab* *les x2, . .,.xk. We get X X Sq1(R) = xi1Sq1(Ri) + xi+11Ri 0 i 2n 0iio2ndd X = Sq1(R0) + xi1Sq1(Ri) 0iio2ndd X + xi+11[Sq1(Ri+1) + Ri]. 0iio2ndd 5032 NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM Since Sq1(R) = 0, we have Sq1(R0) = 0 and Sq1(Ri+1) = Rifor 0 i 2n, i odd. Therefore, X X R = xi1Ri+ xi1Sq1(Ri+1) 0iie2nven 0iio2ndd X = R0 + [xi+11Ri+1+ xi1Sq1(Ri+1)] 0iio2ndd X = R0 + Sq1( xi1Ri+1) 0iio2ndd R0 (mod I0) 0 (mod I0) (by the inductive hypothesis). The lemma is proved. |___| This lemma immediately implies that if all monomials of R 2 Pk are of positive degree, then R2 0 (mod I0). Corollary 2.6.Let k > 1 and suppose S is a non-empty subset of {0, . .,.k - 1} such that 1 62 S. Then QR2 0 (mod I0), Q where Q = s2SQs and R is an arbitrary polynomial in Pk. Proof.As k > 1 and 1 62 S, one gets Sq1(Q) = 0. This implies Sq1(QR2) = 0.__ Thus QR2 0 (mod I0) by Lemma 2.5. The corollary is proved. |__| 3. Proof of the Main Theorem Let Q be a non-zero Dickson monomial. If Q 6= 1, it can be written as Y i Q = A2i, 0 i n where n is some non-negative integer and Ai is some Dickson monomial dividing Qs for i = 0, . .,.n with An 6= 1. 0 s 2 and suppose R is an arbitrary polynomial in Pk. (a) If Q = A2ii6= 1 and it is not full, then QR2n+12 A+ . Pk. 0 i n i m+n+1 n+1 (b) If Q = A2iis full, then QSq2 (R2 ) 2 A+ .Pk for 0 m < k -1. 0 i n Lemma B. Suppose k > 2. If A is a full based cut, then A 0 (mod I1). Proof of the Main Theorem.Suppose Q = A2iiis a Dickson monomial with 0 i n An 6= 1. If Q is not full, then applying Lemma A(a) with R = 1, one gets Q 2 A+ . Pk. If Q is full and n = 0, then Q is the full based cut of itself. So using Lemm* *a B, one obtains Q 0 (mod I1). In particular, Q 2 A+ . Pk. If Q is full and n > 0, then An is the full based cut of itself. By LemmaiB, * *one has An = Sq1(R1) + Sq2(R2), with some R1, R2 2 Pk. Noting that Q0= A2i 0 i 2, 0 m < k - 1 and 0 < j 2m . LetnQ+be1a full Dickson monomialiof heightin and B any Dickson monomial of Sq2 j(Q). Suppose B = B2i, with B2i the ith cut of B and 0 i p Bp 6= 1. We have (a) p n, 5034 NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM (b) If B0= B2ii6= 1, then it is not full. 0 i n Proof.(a) Suppose to the contrary that p < n. We get Y i degQ + 2n+1j = deg( B2i) 0 i p X Y ( 2i) deg( Qs) 0 i p 0 s degQ X Y ( 2i) deg( Qs) (since Q is full) 0 i n 0 (2n+1 - 1) deg( Qs), 0 s 2n deg( Qs), Y 0 deg( Qs). 0 2. This contradiction shows part (a). (b) Suppose to the contrary that B2iiis full. Then 0 i n Y i degQ + 2n+1j = deg( B2i) 0 i p Y i deg( B2i) (mod 2n+1), 0 i n Y i degQ - deg( B2i) 0 (mod 2n+1), 0 i n X 2i(degAi- degBi) 0 (mod 2n+1). 0 i n It is easy to see that degAi- degBi= "idegQ0, with "i2 {0, 1, -1}. Further- more, if "i = 0, then Ai = Bi. So 2i"idegQ0 0 (mod 2n+1). It should be 0 i n noted that degQ0 = 2k - 1 has no common divisor with 2n+1. So 2i"i 0 0 i n (mod 2n+1). This implies "i = 0 for i = 0, . .,.n. In other words, Ai = Bi for THE HIT PROBLEM FOR THE DICKSON ALGEBRA 5035 i = 0, . .,.n and Q = B2ii. We have 0 i n Y i Y i degQ + 2n+1j = deg( B2i) + deg( B2i) 0 i n n 0, we get degB = degQ + 2n+1j > degQ, so p > n. Hence p 2n+1 2n+1 n+1 k-1 deg B2p degBp degQk-1 = 2 . 2 . It implies j 2k-1. Combining this and the fact 2k-1 > 2m j, we obtain j > j. This contradiction comes from the hypothesis that B0is full. Therefore,_the_lem* *ma is proved. |__| Lemma 4.2. Let A 6= 1 be an unfull based cut. Denote by s the smallest integer s 1 such that Qs 6 | A. If s > 1, then there exists for every R 2 Pk an expan* *sion s-1 X 2 AR2 = Sq2 (R1) + BR2 , where R1 2 Pk, every R2 2 Pk and every B is a Dickson monomial with B | Qr, B 6= 1, Qs-1 6 | B. 0 r 2r. Since ar 2s-1 < 2k-1, we have 2r < ar < 2k-1. So, by Theorem 2.1, Sqar(Qr) = 0. Hence Sqar(Qr) = 0. This is true for every (ar)0 0. Thus, the 0th cut C0 of every term* * in Sq2j( Qr) is not divisible by Q0. Recall that ~Ais a divisor of QrQ0. 0 n, so n+1 i-1 n i 0 i n BR2j = ( B2i R2j)2 0 (mod I0). If B2i6= 1, then it is not full by n 1 and the assertion holds for s-1. Then Aq 6=* * 1. By Lemma 4.2, we get n+1 2 2s-1 X 2 QR2 = AqR~ = Sq (R1) + BR2 . Since Qs-1 6 | B, by the inductive hypothesis on s, we have BR222 A+ . Pk. This* * is true for every term BR22, so QR2n+12 A+ . Pk. We now prove Step 2 by induction on n. For n = 0, we have q(Q) = 0, so Lemma A(a) is true by the above remark. Suppose n > 1 and Lemma A(a) holds for every smaller value of n. Using the above remark, it suffices to consider t* *he case q = q(Q) > 0. Again, the proof proceeds by induction on s. For s = 1, we have seen that AqR~2 0 (mod I0). In other words, AqR~2= Sq1(R1) for some R1 2 Pk. Then n+1 Y 2i 2 2q Y 2i 2q 2q QR2 = Ai (AqR~) = Ai Sq (R1 ). 0 i 1 and the assertion holds for every smaller value of s. Since Aq * *6= 1, by Lemma 4.2, we get AqR~2= Sq2s-1(R1) + BR22. So n+1 Y 2i 2 2q QR2 = Ai (AqR~) 0 i 2, it suffices to show Q2Q1 0 (mod I1). From [7, Theorem 2.2], we get X Q1 = xff11.x.f.fkk ff1+...+ffk=2k-2,ff X i=0 or power2of k-1 X 2 2 2 8 2k-1 2 = x1x2x43. .x.2k + x1x2x3x4. .x.k + R , sym sym P where denotes the sum of all symmetrized terms in x1, . .,.xk, and R is some sym polynomial, whose monomials are all of positive degree. By Lemma 2.5, R2 0 (mod I0). We obtain X k-1 k-1 Q1 (x1x2x43. .x.2k + x21x22x23x84. .x.2k ) (mod I0) sym X k-1 Sq2( x1x2x23x84. .x.2k ) (mod I0) sym X k-1 Sq2Sq1( x1x2x3x84. .x.2k ) (mod I0) sym Sq2Sq1(R1) (mod I0), P k-1 where R1 := x1x2x3x84. .x.2k . Writing Q1 = Sq2Sq1(R1) + Sq1(R2) for some sym R2 2 Pk, we get Q2Q1 = Q2Sq2Sq1(R1) + Q2Sq1(R2) R1Sq1Sq2(Q2) + R2Sq1(Q2) (mod I1) (by Lemma 2.4) R1Q0 (mod I1) (by Corollary 2.2). On the other hand, by [7, Theorem 2.2], we have X k-1 X k-1 Q0 = x1x22x43. .x.2k = Sq2( x1x22x23x84. .x.2k ) sym sym X k-1 = Sq2Sq2( x1x2x3x84. .x.2k ) = Sq2Sq2(R1). sym Therefore, Q2Q1 R1Q0 (mod I1) R1Sq2Sq2(R1) (mod I1) Sq2(R1)Sq2(R1) (mod I1) (by Lemma 2.4(b)) [Sq2(R1)]2 (mod I1) 0 (mod I1). Lemma B is proved. |___| References [1]L. E. Dickson, A fundamental system of invariants of the general modular li* *near group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75-98. CM* *P 95:18 [2]Nguy^e~n H. V. Hu'ng, The action of the Steenrod squares on the modular inv* *ariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 1097-1104. MR 92c:55018 5040 NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM [3]Nguy^e~n H. V. Hu'ng, Spherical classes and the algebraic transfer, Trans. * *Amer. Math. Soc. 349 (1997), 3893-3910. MR 98e:55020 [4]Nguy^e~n H. V. Hu'ng, The weak conjecture on spherical classes, Math. Zeit.* * 231 (1999), 727-743. MR 2000g:55019 [5]Nguy^e~n H. V. Hu'ng, Spherical classes and the lambda algebra, Trans. Amer* *. Math. Soc. 353 (2001), 4447-4460. [6]Nguy^e~n H. V. Hu'ng and F. P. Peterson, A-generators for the Dickson algeb* *ra, Trans. Amer. Math. Soc. 347 (1995), 4687-4728. MR 96c:55022 [7]Nguy^e~n H. V. Hu'ng and F. P. Peterson, Spherical classes and the Dickson * *algebra, Math. Proc. Camb. Phil. Soc. 124 (1998), 253-264. MR 99i:55021 [8]M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns* * Hopkins Uni- versity 1990. [9]F. P. Peterson, Generators of H*(RP 1 ^ RP 1) as a module over the Steenrod* * algebra, Abstracts Amer. Math. Soc., No 833, April 1987. [10]S. Priddy, On characterizing summands in the classifying space of a group, * *I, Amer. Jour. Math. 112 (1990), 737-748. MR 91i:55020 [11]J. H. Silverman, Hit polynomials and the canonical antiautomorphism of the * *Steenrod algabra, Proc. Amer. Math. Soc. 123 (1995), 627-637. MR 95c:55023 [12]W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), * *493-523. MR 90i:55035 [13]N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. S* *tudies, No. 50, Princeton Univ. Press, 1962. MR 26:3056 [14]R. M. W. Wood, Modular representations of GL(n, Fp) and homotopy theory, Le* *cture Notes in Math. 1172, Springer Verlag (1985), 188-203. MR 88a:55007 Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr* *~ai Street, Hanoi, Vietnam E-mail address: nhvhung@hotmail.com Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr* *~ai Street, Hanoi, Vietnam E-mail address: trngnam@hotmail.com